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First thing would be to 301 redirect old to new so the new pages have the chance to rank. If you don't you might also run into keyword cannibalisation issues where both old and new try to rank for the same keywords.

In Magento I believe if you disable the old category, it will also be removed the in sitemap.xml it generates for you.

If you're generating the sitemap manually then yes definitely remove them from the sitemap after the redirects.

Is there a reason you need to keep those old categories? In magento you can rename the category and the URL into your new category and it will automatically 301 redirect it to whatever new category URL structure you give it, passing SEO value along.

In the case where a product can fit under many categories/sub-cats and there's a large number of products, it's better to have the product sit under the domain OR a dedicated product folder (/products/product-name). You're still able to link to this single product from many categories and you'll get a nicer looking URL structure.

Magento has the option built in - System > Config > Catalog > SEO

Not sure about worldwide, I'm based in Australia. I just checked my Analytics and it says United States Only.